The four words that do most of the damage
Four terms appear across these institutions' websites and are routinely read as meaning more than they do. Getting them straight resolves most of the confusion this table exists to address.
Licence. Permission from a state or national body to operate as an educational institution in that place. It is about legality, not academic standing. Florida's statute puts it beyond argument: "The granting of a license is not an accreditation." Malta's MFHEA licence is a stronger instrument than Florida's because MFHEA is a higher education regulator rather than a general commercial one — but it is still a provider licence, and provider licensing and programme accreditation are separate steps.
Accreditation. An assessment of an institution or a programme against published standards. The word does real work only when you also know who accredited, and whether that accreditor is itself recognised. QAHE accreditation and regional accreditation in the United States are both called accreditation and are not comparable things.
Candidacy. A process that has started and has not finished. Dunster holds ACBSP and AACSB candidacy; EIMT holds ACBSP candidacy and says in its own words that this means "not yet accredited". Candidacy may become accreditation. It may also lapse. It is not a result, and it should never be presented as one.
Membership. Payment of a subscription to an association. IACBE states this itself: educational membership does not imply accreditation. A membership logo in a footer is a fact about a relationship, not a judgment about quality.
A fifth item belongs here too. An exemption — as in American Management University's exemption from California licensing — is a decision that a body does not need a licence. It is the absence of a requirement, not the presence of an approval.